Introduction to Continuous Symmetries: From Space-Time to Quantum Mechanics

I Symmetry Transformations A Fundamental Symmetries B Symmetries in Classical Mechanics C Symmetries in Quantum Mechanics A_I Euler's and Lagrange's Views in Classical Mechanics 1 Euler's Point of View 2 Lagrange's Point of View II Notions on Group Theory A General Properties of Groups B Linear Representations of a Group A_II Residual Classes of a Subgroup; Quotient Group 1 Residual Classes on the Left 2 Quotient Group III Introduction to Continuous Groups and Lie Groups A General Properties B Examples C Galileo and Poincaré Groups A_III Adjoint Representation, Killing Form, Casimir Operator 1 Representation Adjoint to the Lie Algebra 2 Killing Form; Scalar Product and Change of Basis in L 3 Totally Antisymmetric Structure Constants 4 Casimir Operator IV Representations Induced in the State Space A Conditions Imposed on Transformations in the State Space B Wigner's Theorem C Transformations of Observables D Linear Representations in the State Space E Phase Factors and Projective Representations A_IV Finite-Dimensional Unitary Projective Representations of Related Lie Groups 1 Case Where G is Simply Connected 2 Case Where G is P-Connected B_IV Uhlhorn-Wigner Theorem 1 Real Space 2 Complex Space V Representations of the Galileo and Poincaré Groups: Mass, Spin and Energy A Galileo Group B Poincaré Group A_V Some Properties of the Operators S and W_2 1 Operator S 2 Eigenvalues of the Operator W_2 B_V Geometric Displacement Group 1 Reminders: Classical Properties of Displacements 2 Associated Operators in the State Space C_V Clean Lorentz Group 1 Link with the Group SL(2,C) 2 Small Group Associated with a Four-Vector 3 Operator W_2 D_V Space Reflections (Parity) 1 Action in Real Space 2 Associated Operator in the State Space 3 Retention of Parity VI Construction of State Spaces and Wave Equations A Galileo Group, Schrödinger Equation B Poincaré Group, Klein-Gordon and Dirac Equations A_VI Lagrangians of Wave Equations 1 Lagrangian of a Field 2 Schrödinger's Equation 3 Klein-Gordon Equation 4 Dirac's Equation VII Irreducible Representations of the Group of Rotations, Spinors A Irreducible Unitary Representations of the Group of Rotations B Spin 1/2 Particles; Spinors C Composition of the Kinetic Moments A_VII Homorphism Between SU(2) and Rotation Matrices 1 Transformation of a Vector P Induced by an SU(2) Matrix 2 The Transformation is a Rotation 3 Homomorphism 4 Link to the Reasoning of Chapter VII 5 Link with Bivalent Representations VIII Transformation of Observables by Rotation A Vector Operators B Tensor Operators C Wigner-Eckart Theorem D Decomposition of the Density Matrix on Tensor Operators A_VIII Basic Reminders on Classical Tensors 1 Vectors 2 Tensors 3 Properties 4 Tensoriality Criterion 5 Symmetric and Antisymmetric Tensors 6 Special Tensors 7 Irreducible Tensors B_VIII Second Order Tensor Operators 1 Tensor Product of Two Vector Operators 2 Cartesian Components of the Tensor in the General Case C_VIII Multipolar Moments 1 Electrical Multipole Moments 2 Magnetic Multipole Moments 3 Multipole Moments of a Quantum System for a Given Kinetic Moment Multiplicity J IX Groups SU(2) and SU(3) A System of Discernible but Equivalent Particles B SU(2) Group and Isospin Symmetry C Symmetry SU(3) A_IX the Nature of a Particle Is Equivalent to an Internal Quantum Number 1 Partial or Total Antisymmetrization of a State Vector 2 Correspondence Between the States of Two Physical Systems 3 Physical Consequences B_IX Operators Changing the Symmetry of a State Vector by Permutation 1 Fermions 2 Bosons X Symmetry Breaking A Magnetism, Breaking of the Rotation Symmetry B Some Other Examples APPENDIX I The Reversal of Time 1 Time Reversal in Classical Mechanics 2 Antilinear and Antiunitary Operators in Quantum Mechanics 3 Time Reversal and Antilinearity 4 Explicit Form of the Time Reversal Operator 5 Applications